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Theorem alimp-no-surprise 42527
Description: There is no "surprise" in a for-all with implication if there exists a value where the antecedent is true. This is one way to prevent for-all with implication from allowing anything. For a contrast, see alimp-surprise 42526. The allsome quantifier also counters this problem, see df-alsi 42534. (Contributed by David A. Wheeler, 27-Oct-2018.)
Assertion
Ref Expression
alimp-no-surprise |- -. ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph )
Proof of Theorem alimp-no-surprise
StepHypRef Expression
1 pm4.82 969 . . . . 5 |- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ph )
21albii 1747 . . . 4 |- ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> A. x -. ph )
3 alnex 1706 . . . 4 |- ( A. x -. ph <-> -. E. x ph )
42, 3sylbb 209 . . 3 |- ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) -> -. E. x ph )
5 imnan 438 . . 3 |- ( ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) -> -. E. x ph ) <-> -. ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) /\ E. x ph ) )
64, 5mpbi 220 . 2 |- -. ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) /\ E. x ph )
7 19.26 1798 . . . 4 |- ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) ) )
87anbi2ci 732 . . 3 |- ( ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) /\ E. x ph ) <-> ( E. x ph /\ ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) ) ) )
9 3anass 1042 . . 3 |- ( ( E. x ph /\ A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) ) <-> ( E. x ph /\ ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) ) ) )
10 3anrot 1043 . . 3 |- ( ( E. x ph /\ A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph ) )
118, 9, 103bitr2i 288 . 2 |- ( ( A. x ( ( ph -> ps ) /\ ( ph -> -. ps ) ) /\ E. x ph ) <-> ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph ) )
126, 11mtbi 312 1 |- -. ( A. x ( ph -> ps ) /\ A. x ( ph -> -. ps ) /\ E. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   E.wex 1704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737
This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705
This theorem is referenced by:  alsi-no-surprise  42542
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